For the first time in the United States, a woman who had a uterus transplant has given birth.

The mother, who was born without a uterus, received the transplant from a living donor last year at Baylor University Medical Center in Dallas, and had a baby boy there last month, the hospital said on Friday.

At the family’s request, their name, hometown and the date of the birth are being withheld to protect their privacy, according to Julie Smith, a spokeswoman for the hospital, which is part of Baylor Scott & White Health.

Since 2014, eight other babies have been born to women who had uterus transplants, all in Sweden, at the Sahlgrenska University Hospital in Gothenburg.

A new frontier, uterus transplants are seen as a source of hope for women who cannot give birth because they were born without a uterus or had to have it removed because of cancer, other illness or complications from childbirth. Researchers estimate that in the United States, 50,000 women might be candidates.

The transplants are meant to be temporary, left in place just long enough for a woman to have one or two children, and then removed so she can stop taking the immune-suppressing drugs needed to prevent organ rejection.

Dr. Liza Johannesson, a uterus transplant surgeon who left the Swedish team to join Baylor’s group, said the birth in Dallas was particularly important because it showed that success was not limited to the hospital in Gothenburg.

“To make the field grow and expand and have the procedure come out to more women, it has to be reproduced,” she said, adding that within hours of Baylor’s announcement, advocacy groups for women with uterine infertility from all over the world had contacted her to express their excitement at the news.

“It was a very exciting birth,” Dr. Johannesson said. “I’ve seen so many births and delivered so many babies, but this was a very special one.”

At Baylor, eight women have had transplants, including the new mother, in a clinical trial designed to include 10 patients. One recipient is pregnant, and two others — one of whom received her transplant from a deceased donor — are trying to conceive. Four other transplants failed after the surgery, and the organs had to be removed, said Dr. Giuliano Testa, principal investigator of the research project and surgical chief of abdominal transplantation.

“We had a very rough start, and then hit the right path,” Dr. Testa said in a telephone interview. “Who paid for it in a certain way were the first three women. I feel very thankful for their contribution, more so than I can express.”

Both Dr. Johannesson and Dr. Testa said that a large part of their motivation came from meeting patients and coming to understand how devastated they were to find out that they would not be able to have children.

Dr. Testa said: “I think many men will never understand this fully, to understand the desire of these women to be mothers. What moved all of us is to see the mother holding her baby, when she was told, ‘You will never have it.’”

The transplants are now experimental, with much of the cost covered by research funds. But they are expensive, and if they become part of medical practice, will probably cost hundreds of thousands of dollars. It is not clear that insurers will pay, and Dr. Testa acknowledged that many women who want the surgery will not be able to afford it.

Another hospital, the Cleveland Clinic, performed the first uterus transplant in the United States in February 2016, but it failed after two weeks because of an infection that caused life-threatening hemorrhage and required emergency surgery to remove the organ. The clinic halted its program for an extended period, but has restarted it and has patients awaiting transplants, a spokeswoman, Victoria Vinci, said.

The woman who gave birth at Baylor was the fourth to receive a transplant there, in September 2016.

The process is complicated and has considerable risks for both recipients and donors. Donors undergo a five-hour operation that is more complex and takes out more tissue than a standard hysterectomy to remove the uterus. The transplant surgery is also difficult, in some ways comparable to a liver transplant, Dr. Testa said.

Recipients face the risks of surgery and anti-rejection drugs for a transplant that they, unlike someone with heart or liver failure, do not need to save their lives. Their pregnancies are considered high-risk, and the babies have to be delivered by cesarean section to avoid putting too much strain on the transplanted uterus. So far all the births have occurred a bit earlier than the normal 40 weeks of gestation — at 32 to 36 weeks.

Women who have transplants cannot conceive the natural way, because their ovaries are not connected to the uterus, so there is no way for an egg to get in there. Instead, they need in vitro fertilization. Before the transplant, women are given hormone treatments to make their ovaries release multiple eggs, which are then harvested, fertilized and frozen.

Once the woman has fully recovered from surgery and begun menstruating, the eggs can be implanted in the uterus, one at a time, until she becomes pregnant.

In Sweden, doctors waited a year after the transplant before trying to start a pregnancy, to allow the women time to heal. At Baylor, the team moved much faster, and began trying to impregnate the women within a few months of the surgery, soon after they began menstruating.

Dr. Testa said it was his idea to start the pregnancies earlier, because the women were young and healthy, and did not need a year to bounce back from surgery. He argued that the waiting time just kept them on anti-rejection drugs — which have significant side effects — for longer than necessary.

“We went shorter,” he said. “I think we were right.”

He and Dr. Johannesson said the Swedish team, and other centers planning transplants, had also begun to consider shortening the wait.